Electrical Engineering
      and Computer Sciences

Electrical Engineering and Computer Sciences

COLLEGE OF ENGINEERING

UC Berkeley

Geometric Continuity of Parametric Curves

Brian A. Barsky and Anthony D. DeRose

EECS Department
University of California, Berkeley
Technical Report No. UCB/CSD-84-205
October 1984

http://www.eecs.berkeley.edu/Pubs/TechRpts/1984/CSD-84-205.pdf

Parametric spline curves are typically constructed so that the first n parametric derivatives agree where the curve segments abut. This type of continuity condition has become known as C^n or nth order parametric continuity. We show that the use of parametric continuity disallows many parametrizations which generate geometrically smooth curves.

We define nth order geometric continuity (G^n), develop constraint equations that are necessary and sufficient for geometric continuity of curves, and show that geometric continuity is a relaxed form of parametric continuity. G^n continuity provides for the introduction of n quantities known as shape parameters which can be made available to a designer in a computer aided design environment to modify the shape of curves without moving control vertices. Several applications of the theory are discussed, along with topics of future research.


BibTeX citation:

@techreport{Barsky:CSD-84-205,
    Author = {Barsky, Brian A. and DeRose, Anthony D.},
    Title = {Geometric Continuity of Parametric Curves},
    Institution = {EECS Department, University of California, Berkeley},
    Year = {1984},
    Month = {Oct},
    URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1984/5752.html},
    Number = {UCB/CSD-84-205},
    Abstract = {Parametric spline curves are typically constructed so that the first <i>n</i> parametric derivatives agree where the curve segments abut. This type of continuity condition has become known as <i>C^n</i> or <i>n</i>th order parametric continuity. We show that the use of parametric continuity disallows many parametrizations which generate geometrically smooth curves.  <p>  We define <i>n</i>th order geometric continuity (<i>G^n</i>), develop constraint equations that are necessary and sufficient for geometric continuity of curves, and show that geometric continuity is a relaxed form of parametric continuity. <i>G^n</i> continuity provides for the introduction of <i>n</i> quantities known as shape parameters which can be made available to a designer in a computer aided design environment to modify the shape of curves without moving control vertices.  Several applications of the theory are discussed, along with topics of future research.}
}

EndNote citation:

%0 Report
%A Barsky, Brian A.
%A DeRose, Anthony D.
%T Geometric Continuity of Parametric Curves
%I EECS Department, University of California, Berkeley
%D 1984
%@ UCB/CSD-84-205
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/1984/5752.html
%F Barsky:CSD-84-205